Problem: A circular dartboard is divided into regions with various central angles, as shown. The probability of a dart randomly landing in a particular region is $\frac16$. What is the corresponding measure, in degrees, of the central angle of this section of the dartboard? [asy]
unitsize(1.5cm);
defaultpen(linewidth(.7pt));

pair O=(0,0);
draw(Circle(O,1));
draw(dir(0)--O--dir(90));
draw(dir(150)--O--dir(225));
[/asy]
Let $A$ be the area of the circular dartboard.  If the measure of a central angle of a sector is $x$ degrees, then the area of the sector is $\left(\frac{x}{360}\right)A$.  The probability of the dart landing in a region is ratio of the area of the region to the area of the dartboard, so  \[
\frac{1}{6} = \frac{\left(\frac{x}{360}\right)A}{A}.
\] Solve to find $x=\boxed{60}$.